> Date: Tue, 31 Oct 2000 08:17:03 +0100
> From: Torkel Franzen <torkel at sm.luth.se>
> that a purported observation such as
>> (1) Even if Goldbach's conjecture is true, it is not necessarily
> provable in ZFC
>> is mysterious. In what way is (1) mysterious?
Misterious is not a proper word, fraudulent would fit better.
To be scientifically considerable, "thesis" (1) has to be
preceded by at least explanation, if not a rigorous definition,
what is the intended meaning of "true". That has not been made
clear in the course of the discussion.
I can see three possible ways to specify the meaning of
the phrase Goldbach's conjecture is true:
(a) it has been correctly proved mathematically
(b) it is true as a fact of the nature
(c) that it is true is given in a sacred scripts
and there seems to be no other way to understand it modulo
variations. As (a) makes "thesis" (1) false and (c) leads
us out of the science, only (b) remains.
Thus you say "Goldbach's conjecture is true" meaning that
any physical amount of "pebbles" or whatever counting units
is taken it never yields a counterexample to the conjecture.
Note that, in this argument, the "pebbles" must be real
physical ones, not philosophical, imaginary counting units,
to retain the empirically valid character of (b).
But having this accepted, we immediately face the failure
because (according to modern physics)
the universe has a finite number of particles.
Moreover, if you pretend to have the predicate "n is prime"
as physically valid you should give some counting method,
some actual way at least IN PRINCIPLE to see if n is prime.
Now you will have another bump: quantum mechanics says that
any counting method will give mistakes with certain
probability (0 when you count your today's gain at Wall St,
but very much non-0 when you arrive to astronomical amounts),
even worse, your "pebbles" themselves will disappear with
time due to quantum effects,
so, if you really want to go with (b) you have first
to convince us that Goldbach's conjecture is really an
observable, physically valid statement, only then it is
possible to consider it to be "true" if, by definition, there
is no any (and never be) empirical counterexample.
V.Kanovei